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Flat knot 6.1668

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,0,1,2,1,1,0,1,1,0,0,0,-1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1668', '7.37533']
Arrow polynomial of the knot is: 4K13 - 14K1*2 - 4K1K2 - K1 + 7K2 + K3 + 8
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1023', '6.1530', '6.1662', '6.1668', '6.1801']
Outer characteristic polynomial of the knot is: t^7+34t^5+56t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1668', '7.37533']
2-strand cable arrow polynomial of the knot is: -1024K16 - 1472K1*4K2*2 + 6048K1*4K2 - 8752K14 + 1376K1*3K2K3 - 1344K1*3K3 - 576K12K2*4 + 4928K1*2K2*3 + 128K1*2K2*2K4 - 15872K12K2*2 - 1088K1*2K2K4 + 12216K1*2K2 - 624K12K3*2 - 316K1*2 + 992K1K23K3 - 2272K1K2*2K3 - 256K1K2*2K5 - 160K1K2K3K4 + 8664K1K2K3 + 528K1K3K4 + 24K1K4K5 - 32K2*6 + 64K2*4K4 - 3464K24 - 32K2*3K6 - 496K22K3*2 - 16K2*2K4*2 + 2272K2*2K4 - 1294K22 + 288K2K3K5 + 16K2K4K6 - 908K3*2 - 222K4*2 - 32K5*2 - 2K6**2 + 2708
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1668']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.59', 'vk6.114', 'vk6.209', 'vk6.256', 'vk6.299', 'vk6.681', 'vk6.1219', 'vk6.1266', 'vk6.1355', 'vk6.1402', 'vk6.1449', 'vk6.1933', 'vk6.2383', 'vk6.2443', 'vk6.2937', 'vk6.2993', 'vk6.5732', 'vk6.5763', 'vk6.7797', 'vk6.7828', 'vk6.13283', 'vk6.13316', 'vk6.14774', 'vk6.14814', 'vk6.15930', 'vk6.15970', 'vk6.18055', 'vk6.24493', 'vk6.33032', 'vk6.33384', 'vk6.43921', 'vk6.50494']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is c.
The reverse -K is
The mirror image K* is
The reversed mirror image -K* is
The fillings (up to the first 10) associated to the algebraic genus:
Or click here to check the fillings

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,0,1,2,1,1,0,1,1,0,0,0,-1,-1]

Flat knots (up to 7 crossings) with same phi are :['6.1668', '7.37533']

Arrow polynomial of the knot is: 4*K1**3 - 14*K1**2 - 4*K1*K2 - K1 + 7*K2 + K3 + 8

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1023', '6.1530', '6.1662', '6.1668', '6.1801']

Outer characteristic polynomial of the knot is: t^7+34t^5+56t^3+9t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1668', '7.37533']

2-strand cable arrow polynomial of the knot is: -1024*K1**6 - 1472*K1**4*K2**2 + 6048*K1**4*K2 - 8752*K1**4 + 1376*K1**3*K2*K3 - 1344*K1**3*K3 - 576*K1**2*K2**4 + 4928*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 15872*K1**2*K2**2 - 1088*K1**2*K2*K4 + 12216*K1**2*K2 - 624*K1**2*K3**2 - 316*K1**2 + 992*K1*K2**3*K3 - 2272*K1*K2**2*K3 - 256*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 8664*K1*K2*K3 + 528*K1*K3*K4 + 24*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 3464*K2**4 - 32*K2**3*K6 - 496*K2**2*K3**2 - 16*K2**2*K4**2 + 2272*K2**2*K4 - 1294*K2**2 + 288*K2*K3*K5 + 16*K2*K4*K6 - 908*K3**2 - 222*K4**2 - 32*K5**2 - 2*K6**2 + 2708

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1668']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.59', 'vk6.114', 'vk6.209', 'vk6.256', 'vk6.299', 'vk6.681', 'vk6.1219', 'vk6.1266', 'vk6.1355', 'vk6.1402', 'vk6.1449', 'vk6.1933', 'vk6.2383', 'vk6.2443', 'vk6.2937', 'vk6.2993', 'vk6.5732', 'vk6.5763', 'vk6.7797', 'vk6.7828', 'vk6.13283', 'vk6.13316', 'vk6.14774', 'vk6.14814', 'vk6.15930', 'vk6.15970', 'vk6.18055', 'vk6.24493', 'vk6.33032', 'vk6.33384', 'vk6.43921', 'vk6.50494']

The R3 orbit of minmal crossing diagrams contains:

The diagrammatic symmetry type of this knot is c.

The reverse -K is

The mirror image K* is

The reversed mirror image -K* is

The fillings (up to the first 10) associated to the algebraic genus:

Or click here to check the fillings

invariant	value
Gauss code	O1O2O3U2O4U3U5U6O5O6U1U4
R3 orbit	{'O1O2O3U2O4U3U5U6O5O6U1U4'}
R3 orbit length	1
Gauss code of -K	O1O2O3U4U3O5O6U5U6U1O4U2
Gauss code of K*	O1O2O3U2U3O4O5U4U6U1O6U5
Gauss code of -K*	O1O2O3U4O5U3U5U6O4O6U1U2
Diagrammatic symmetry type	c
Flat genus of the diagram	3
If K is checkerboard colorable	False
If K is almost classical	False
Based matrix from Gauss code	[[ 0 -1 -1 0 2 -1 1],[ 1 0 -1 1 2 0 2],[ 1 1 0 1 1 1 1],[ 0 -1 -1 0 1 0 0],[-2 -2 -1 -1 0 -3 -1],[ 1 0 -1 0 3 0 1],[-1 -2 -1 0 1 -1 0]]
Primitive based matrix	[[ 0 2 1 0 -1 -1 -1],[-2 0 -1 -1 -1 -2 -3],[-1 1 0 0 -1 -2 -1],[ 0 1 0 0 -1 -1 0],[ 1 1 1 1 0 1 1],[ 1 2 2 1 -1 0 0],[ 1 3 1 0 -1 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-1,0,1,1,1,1,1,1,2,3,0,1,2,1,1,1,0,-1,-1,0]
Phi over symmetry	[-2,-1,0,1,1,1,0,1,0,1,2,1,1,0,1,1,0,0,0,-1,-1]
Phi of -K	[-1,-1,-1,0,1,2,-1,-1,0,1,2,0,0,0,1,1,1,0,1,1,0]
Phi of K*	[-2,-1,0,1,1,1,0,1,0,1,2,1,1,0,1,1,0,0,0,-1,-1]
Phi of -K*	[-1,-1,-1,0,1,2,-1,0,0,1,3,1,1,1,1,1,2,2,0,1,1]
Symmetry type of based matrix	c
u-polynomial	-t^2+2t
Normalized Jones-Krushkal polynomial	3z^2+22z+33
Enhanced Jones-Krushkal polynomial	3w^3z^2+22w^2z+33w
Inner characteristic polynomial	t^6+26t^4+39t^2+4
Outer characteristic polynomial	t^7+34t^5+56t^3+9t
Flat arrow polynomial	4K13 - 14K1*2 - 4K1K2 - K1 + 7K2 + K3 + 8
2-strand cable arrow polynomial	-1024K16 - 1472K1*4K2*2 + 6048K1*4K2 - 8752K14 + 1376K1*3K2K3 - 1344K1*3K3 - 576K12K2*4 + 4928K1*2K2*3 + 128K1*2K2*2K4 - 15872K12K2*2 - 1088K1*2K2K4 + 12216K1*2K2 - 624K12K3*2 - 316K1*2 + 992K1K23K3 - 2272K1K2*2K3 - 256K1K2*2K5 - 160K1K2K3K4 + 8664K1K2K3 + 528K1K3K4 + 24K1K4K5 - 32K2*6 + 64K2*4K4 - 3464K24 - 32K2*3K6 - 496K22K3*2 - 16K2*2K4*2 + 2272K2*2K4 - 1294K22 + 288K2K3K5 + 16K2K4K6 - 908K3*2 - 222K4*2 - 32K5*2 - 2K6**2 + 2708
Genus of based matrix	1
Fillings of based matrix	[[{3, 6}, {2, 5}, {1, 4}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {4}, {2, 3}, {1}]]
If K is slice	False

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